A note on the polynomial moments of the partition function in the SK model

TL;DR

This paper establishes a new identity linking the moments of the partition function in the SK model, providing an alternative characterization of its asymptotic behavior as the system size grows.

Contribution

It introduces a simple identity connecting the moments of the partition function for different system sizes, offering a new perspective on the asymptotic analysis in the SK model.

Findings

Derived a relation between the $k$th and $N$th moments of the partition function.

Provided an alternative characterization of the limit of the scaled log-moments.

Simplified the understanding of the asymptotic behavior of the partition function in the SK model.

Abstract

We prove a simple identity relating the $k$th moment of the partition function $Z_N(\cdot)$ in the SK model to the $N$th moment of the partition function $Z_k(\cdot)$. As a corollary we find a characterisation of the limit $\lim_{N \to \infty} \frac{1}{N} \log \mathbb{E} Z_N(\beta)^k$ alternative to the one found previously by Michel Talagrand in \cite{4}.